Random equations in nilpotent groups

In this paper we study satisfiability of random equations in an infinite finitely generated nilpotent group G. Let Sat(G,k) be the set of all satisfiable equations over G in k variables. For a free abelian group Am of rank m we show that the asymptotic density ρ(Sat(Am,k)) of the set Sat(Am,k) (in the whole space of all equations in k variables over G) is equal to 0 for k=1, and is equal to ζ(k+m)/ζ(k) for k⩾2. More generally, if G is a finitely generated nilpotent infinite group, then again the asymptotic density of the set Sat(G,1) is 0. For k⩾2 we give robust estimates for the upper and lower asymptotic densities of the set Sat(G,k). Namely, we prove that these densities lie in the interval from to , where h(G) is the Hirsch length of G, t(G) is the order of lower central torsion of G, and m is the torsion-free rank (Hirsch length) of the abelianization of G. This allows one to describe the asymptotic behavior of the set Sat(G,k) for different parameters k,m,h(G),t(G).